Rainbowfish need a GH above 150ppm and pH above 7.0.
Your tetras come from soft water with a GH below 100ppm nd pH below 7.0.
Big rainbowfish will bully the smaller tetras and get all the food. I have also seen big rainbows go after Otocinclus catfish and small bristlenose catfish.
If you have a good sized tank (3 foot or longer), then you can keep smaller species of rainbowfish with tetras as long as you keep the GH around 150ppm and the pH around 7.0.
Rainbowfish do best in groups of at least 6 and preferably more. However, they will school with other species of rainbowfish so if you had 3 Melanotaenia praecox and 3 M. maccullochi they would show off to each other and hang around together. However, they do prefer to be kept with their own kind.
A lot of people breed rainbowfish in trios (1 male & 2 females) and they are often sold in trios for this reason.
Rainbowfish need lots of plant matter in their diet and are sensitive to chemicals. When you do water changes, make sure any new water is free of chlorine/ chloramine or you can poison and kill the fish.
The following link has information about rainbowfish.
A website dedicated to Rainbowfishes of Australia and New Guinea
rainbowfish.angfaqld.org.au